If the question really is just about redirecting from one domain to a site on another, I have some solutions:
Where you set up a frameset on a page that basically takes up 100% of the page and loads the content from another site into it.
So you type: www.site1.com and it has a simple HTML frame page, which then loads in www.sitwithlongurl.com/evenlongerbit
Advantage - no server access needed other than FTP and the ability to upload a simple page. Users will see your short URL in their browser bar.
Disadvantage - your short URL won't get indexed as the full site, it will just be indexed as one single page of frames; not really ideal (i'd hide it from the index anyway as it will be duplicate content).
Where you set up a page on your domain that simply redirects, ie using PHP:
Advantage - its pretty clean, its transparent. Teleports users directly to the long URL and makes no secret of what its doing.
Disadvantage - You'll need PHP on the hosting server. Fairly common tho.
If you have control of DNS records, you can manipulate them and forward to another site at that level.